(*  Title:      HOL/Proofs/Lambda/LambdaType.thy
    Author:     Stefan Berghofer
    Copyright   2000 TU Muenchen
*)

section \<open>Simply-typed lambda terms\<close>

theory LambdaType imports ListApplication begin


subsection \<open>Environments\<close>

definition
  shift :: "(nat \<Rightarrow> 'a) \<Rightarrow> nat \<Rightarrow> 'a \<Rightarrow> nat \<Rightarrow> 'a"  ("_\<langle>_:_\<rangle>" [90, 0, 0] 91) where
  "e\<langle>i:a\<rangle> = (\<lambda>j. if j < i then e j else if j = i then a else e (j - 1))"

lemma shift_eq [simp]: "i = j \<Longrightarrow> (e\<langle>i:T\<rangle>) j = T"
  by (simp add: shift_def)

lemma shift_gt [simp]: "j < i \<Longrightarrow> (e\<langle>i:T\<rangle>) j = e j"
  by (simp add: shift_def)

lemma shift_lt [simp]: "i < j \<Longrightarrow> (e\<langle>i:T\<rangle>) j = e (j - 1)"
  by (simp add: shift_def)

lemma shift_commute [simp]: "e\<langle>i:U\<rangle>\<langle>0:T\<rangle> = e\<langle>0:T\<rangle>\<langle>Suc i:U\<rangle>"
  by (rule ext) (simp_all add: shift_def split: nat.split)


subsection \<open>Types and typing rules\<close>

datatype type =
    Atom nat
  | Fun type type    (infixr "\<Rightarrow>" 200)

inductive typing :: "(nat \<Rightarrow> type) \<Rightarrow> dB \<Rightarrow> type \<Rightarrow> bool"  ("_ \<turnstile> _ : _" [50, 50, 50] 50)
  where
    Var [intro!]: "env x = T \<Longrightarrow> env \<turnstile> Var x : T"
  | Abs [intro!]: "env\<langle>0:T\<rangle> \<turnstile> t : U \<Longrightarrow> env \<turnstile> Abs t : (T \<Rightarrow> U)"
  | App [intro!]: "env \<turnstile> s : T \<Rightarrow> U \<Longrightarrow> env \<turnstile> t : T \<Longrightarrow> env \<turnstile> (s \<degree> t) : U"

inductive_cases typing_elims [elim!]:
  "e \<turnstile> Var i : T"
  "e \<turnstile> t \<degree> u : T"
  "e \<turnstile> Abs t : T"

primrec
  typings :: "(nat \<Rightarrow> type) \<Rightarrow> dB list \<Rightarrow> type list \<Rightarrow> bool"
where
    "typings e [] Ts = (Ts = [])"
  | "typings e (t # ts) Ts =
      (case Ts of
        [] \<Rightarrow> False
      | T # Ts \<Rightarrow> e \<turnstile> t : T \<and> typings e ts Ts)"

abbreviation
  typings_rel :: "(nat \<Rightarrow> type) \<Rightarrow> dB list \<Rightarrow> type list \<Rightarrow> bool"
    ("_ \<tturnstile> _ : _" [50, 50, 50] 50) where
  "env \<tturnstile> ts : Ts == typings env ts Ts"

abbreviation
  funs :: "type list \<Rightarrow> type \<Rightarrow> type"  (infixr "\<Rrightarrow>" 200) where
  "Ts \<Rrightarrow> T == foldr Fun Ts T"


subsection \<open>Some examples\<close>

schematic_goal "e \<turnstile> Abs (Abs (Abs (Var 1 \<degree> (Var 2 \<degree> Var 1 \<degree> Var 0)))) : ?T"
  by force

schematic_goal "e \<turnstile> Abs (Abs (Abs (Var 2 \<degree> Var 0 \<degree> (Var 1 \<degree> Var 0)))) : ?T"
  by force


subsection \<open>Lists of types\<close>

lemma lists_typings:
    "e \<tturnstile> ts : Ts \<Longrightarrow> listsp (\<lambda>t. \<exists>T. e \<turnstile> t : T) ts"
  apply (induct ts arbitrary: Ts)
   apply (case_tac Ts)
     apply simp
     apply (rule listsp.Nil)
    apply simp
  apply (case_tac Ts)
   apply simp
  apply simp
  apply (rule listsp.Cons)
   apply blast
  apply blast
  done

lemma types_snoc: "e \<tturnstile> ts : Ts \<Longrightarrow> e \<turnstile> t : T \<Longrightarrow> e \<tturnstile> ts @ [t] : Ts @ [T]"
  apply (induct ts arbitrary: Ts)
  apply simp
  apply (case_tac Ts)
  apply simp+
  done

lemma types_snoc_eq: "e \<tturnstile> ts @ [t] : Ts @ [T] =
  (e \<tturnstile> ts : Ts \<and> e \<turnstile> t : T)"
  apply (induct ts arbitrary: Ts)
  apply (case_tac Ts)
  apply simp+
  apply (case_tac Ts)
  apply (case_tac "ts @ [t]")
  apply simp+
  done

lemma rev_exhaust2 [extraction_expand]:
  obtains (Nil) "xs = []"  |  (snoc) ys y where "xs = ys @ [y]"
  \<comment> \<open>Cannot use \<open>rev_exhaust\<close> from the \<open>List\<close>
    theory, since it is not constructive\<close>
  apply (subgoal_tac "\<forall>ys. xs = rev ys \<longrightarrow> thesis")
  apply (erule_tac x="rev xs" in allE)
  apply simp
  apply (rule allI)
  apply (rule impI)
  apply (case_tac ys)
  apply simp
  apply simp
  done

lemma types_snocE: "e \<tturnstile> ts @ [t] : Ts \<Longrightarrow>
  (\<And>Us U. Ts = Us @ [U] \<Longrightarrow> e \<tturnstile> ts : Us \<Longrightarrow> e \<turnstile> t : U \<Longrightarrow> P) \<Longrightarrow> P"
  apply (cases Ts rule: rev_exhaust2)
  apply simp
  apply (case_tac "ts @ [t]")
  apply (simp add: types_snoc_eq)+
  done


subsection \<open>n-ary function types\<close>

lemma list_app_typeD:
    "e \<turnstile> t \<degree>\<degree> ts : T \<Longrightarrow> \<exists>Ts. e \<turnstile> t : Ts \<Rrightarrow> T \<and> e \<tturnstile> ts : Ts"
  apply (induct ts arbitrary: t T)
   apply simp
  apply (rename_tac a b t T)
  apply atomize
  apply simp
  apply (erule_tac x = "t \<degree> a" in allE)
  apply (erule_tac x = T in allE)
  apply (erule impE)
   apply assumption
  apply (elim exE conjE)
  apply (ind_cases "e \<turnstile> t \<degree> u : T" for t u T)
  apply (rule_tac x = "Ta # Ts" in exI)
  apply simp
  done

lemma list_app_typeE:
  "e \<turnstile> t \<degree>\<degree> ts : T \<Longrightarrow> (\<And>Ts. e \<turnstile> t : Ts \<Rrightarrow> T \<Longrightarrow> e \<tturnstile> ts : Ts \<Longrightarrow> C) \<Longrightarrow> C"
  by (insert list_app_typeD) fast

lemma list_app_typeI:
    "e \<turnstile> t : Ts \<Rrightarrow> T \<Longrightarrow> e \<tturnstile> ts : Ts \<Longrightarrow> e \<turnstile> t \<degree>\<degree> ts : T"
  apply (induct ts arbitrary: t T Ts)
   apply simp
  apply (rename_tac a b t T Ts)
  apply atomize
  apply (case_tac Ts)
   apply simp
  apply simp
  apply (erule_tac x = "t \<degree> a" in allE)
  apply (erule_tac x = T in allE)
  apply (rename_tac list)
  apply (erule_tac x = list in allE)
  apply (erule impE)
   apply (erule conjE)
   apply (erule typing.App)
   apply assumption
  apply blast
  done

text \<open>
For the specific case where the head of the term is a variable,
the following theorems allow to infer the types of the arguments
without analyzing the typing derivation. This is crucial
for program extraction.
\<close>

theorem var_app_type_eq:
  "e \<turnstile> Var i \<degree>\<degree> ts : T \<Longrightarrow> e \<turnstile> Var i \<degree>\<degree> ts : U \<Longrightarrow> T = U"
  apply (induct ts arbitrary: T U rule: rev_induct)
  apply simp
  apply (ind_cases "e \<turnstile> Var i : T" for T)
  apply (ind_cases "e \<turnstile> Var i : T" for T)
  apply simp
  apply simp
  apply (ind_cases "e \<turnstile> t \<degree> u : T" for t u T)
  apply (ind_cases "e \<turnstile> t \<degree> u : T" for t u T)
  apply atomize
  apply (erule_tac x="Ta \<Rightarrow> T" in allE)
  apply (erule_tac x="Tb \<Rightarrow> U" in allE)
  apply (erule impE)
  apply assumption
  apply (erule impE)
  apply assumption
  apply simp
  done

lemma var_app_types: "e \<turnstile> Var i \<degree>\<degree> ts \<degree>\<degree> us : T \<Longrightarrow> e \<tturnstile> ts : Ts \<Longrightarrow>
  e \<turnstile> Var i \<degree>\<degree> ts : U \<Longrightarrow> \<exists>Us. U = Us \<Rrightarrow> T \<and> e \<tturnstile> us : Us"
  apply (induct us arbitrary: ts Ts U)
  apply simp
  apply (erule var_app_type_eq)
  apply assumption
  apply simp
  apply (rename_tac a b ts Ts U)
  apply atomize
  apply (case_tac U)
  apply (rule FalseE)
  apply simp
  apply (erule list_app_typeE)
  apply (ind_cases "e \<turnstile> t \<degree> u : T" for t u T)
  apply (rename_tac nat Tsa Ta)
  apply (drule_tac T="Atom nat" and U="Ta \<Rightarrow> Tsa \<Rrightarrow> T" in var_app_type_eq)
  apply assumption
  apply simp
  apply (rename_tac nat type1 type2)
  apply (erule_tac x="ts @ [a]" in allE)
  apply (erule_tac x="Ts @ [type1]" in allE)
  apply (erule_tac x="type2" in allE)
  apply simp
  apply (erule impE)
  apply (rule types_snoc)
  apply assumption
  apply (erule list_app_typeE)
  apply (ind_cases "e \<turnstile> t \<degree> u : T" for t u T)
  apply (drule_tac T="type1 \<Rightarrow> type2" and U="Ta \<Rightarrow> Tsa \<Rrightarrow> T" in var_app_type_eq)
  apply assumption
  apply simp
  apply (erule impE)
  apply (rule typing.App)
  apply assumption
  apply (erule list_app_typeE)
  apply (ind_cases "e \<turnstile> t \<degree> u : T" for t u T)
  apply (frule_tac T="type1 \<Rightarrow> type2" and U="Ta \<Rightarrow> Tsa \<Rrightarrow> T" in var_app_type_eq)
  apply assumption
  apply simp
  apply (erule exE)
  apply (rule_tac x="type1 # Us" in exI)
  apply simp
  apply (erule list_app_typeE)
  apply (ind_cases "e \<turnstile> t \<degree> u : T" for t u T)
  apply (frule_tac T="type1 \<Rightarrow> Us \<Rrightarrow> T" and U="Ta \<Rightarrow> Tsa \<Rrightarrow> T" in var_app_type_eq)
  apply assumption
  apply simp
  done

lemma var_app_typesE: "e \<turnstile> Var i \<degree>\<degree> ts : T \<Longrightarrow>
  (\<And>Ts. e \<turnstile> Var i : Ts \<Rrightarrow> T \<Longrightarrow> e \<tturnstile> ts : Ts \<Longrightarrow> P) \<Longrightarrow> P"
  apply (drule var_app_types [of _ _ "[]", simplified])
  apply (iprover intro: typing.Var)+
  done

lemma abs_typeE: "e \<turnstile> Abs t : T \<Longrightarrow> (\<And>U V. e\<langle>0:U\<rangle> \<turnstile> t : V \<Longrightarrow> P) \<Longrightarrow> P"
  apply (cases T)
  apply (rule FalseE)
  apply (erule typing.cases)
  apply simp_all
  apply atomize
  apply (rename_tac type1 type2)
  apply (erule_tac x="type1" in allE)
  apply (erule_tac x="type2" in allE)
  apply (erule mp)
  apply (erule typing.cases)
  apply simp_all
  done


subsection \<open>Lifting preserves well-typedness\<close>

lemma lift_type [intro!]: "e \<turnstile> t : T \<Longrightarrow> e\<langle>i:U\<rangle> \<turnstile> lift t i : T"
  by (induct arbitrary: i U set: typing) auto

lemma lift_types:
  "e \<tturnstile> ts : Ts \<Longrightarrow> e\<langle>i:U\<rangle> \<tturnstile> (map (\<lambda>t. lift t i) ts) : Ts"
  apply (induct ts arbitrary: Ts)
   apply simp
  apply (case_tac Ts)
   apply auto
  done


subsection \<open>Substitution lemmas\<close>

lemma subst_lemma:
    "e \<turnstile> t : T \<Longrightarrow> e' \<turnstile> u : U \<Longrightarrow> e = e'\<langle>i:U\<rangle> \<Longrightarrow> e' \<turnstile> t[u/i] : T"
  apply (induct arbitrary: e' i U u set: typing)
    apply (rule_tac x = x and y = i in linorder_cases)
      apply auto
  apply blast
  done

lemma substs_lemma:
  "e \<turnstile> u : T \<Longrightarrow> e\<langle>i:T\<rangle> \<tturnstile> ts : Ts \<Longrightarrow>
     e \<tturnstile> (map (\<lambda>t. t[u/i]) ts) : Ts"
  apply (induct ts arbitrary: Ts)
   apply (case_tac Ts)
    apply simp
   apply simp
  apply atomize
  apply (case_tac Ts)
   apply simp
  apply simp
  apply (erule conjE)
  apply (erule (1) subst_lemma)
  apply (rule refl)
  done


subsection \<open>Subject reduction\<close>

lemma subject_reduction: "e \<turnstile> t : T \<Longrightarrow> t \<rightarrow>\<^sub>\<beta> t' \<Longrightarrow> e \<turnstile> t' : T"
  apply (induct arbitrary: t' set: typing)
    apply blast
   apply blast
  apply atomize
  apply (ind_cases "s \<degree> t \<rightarrow>\<^sub>\<beta> t'" for s t t')
    apply hypsubst
    apply (ind_cases "env \<turnstile> Abs t : T \<Rightarrow> U" for env t T U)
    apply (rule subst_lemma)
      apply assumption
     apply assumption
    apply (rule ext)
    apply (case_tac x)
     apply auto
  done

theorem subject_reduction': "t \<rightarrow>\<^sub>\<beta>\<^sup>* t' \<Longrightarrow> e \<turnstile> t : T \<Longrightarrow> e \<turnstile> t' : T"
  by (induct set: rtranclp) (iprover intro: subject_reduction)+


subsection \<open>Alternative induction rule for types\<close>

lemma type_induct [induct type]:
  assumes
  "(\<And>T. (\<And>T1 T2. T = T1 \<Rightarrow> T2 \<Longrightarrow> P T1) \<Longrightarrow>
    (\<And>T1 T2. T = T1 \<Rightarrow> T2 \<Longrightarrow> P T2) \<Longrightarrow> P T)"
  shows "P T"
proof (induct T)
  case Atom
  show ?case by (rule assms) simp_all
next
  case Fun
  show ?case by (rule assms) (insert Fun, simp_all)
qed

end